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 Incoming links: Run of successes

Run of successes Created 2026-09-24 Updated 2026-11-05

A run of successes is a consecutive block of successful trials in a binary observation sequence. For independent trials with success probability

0 < p < 1

, the waiting time

T_{r}

until the first run of

r

successes has mean

E T_{r} = \frac{1 - p ^{r}}{( 1 - p ) p ^{r}} .

Let

E_{k}

be the remaining mean after

k

consecutive successes. For

k < r

E_{k} = 1 + p E_{k + 1} + (1 - p) E_{0}

, while

E_{r} = 0

. Solving backward gives the formula. At

p = 1/2

r = 2

, the mean is six, agreeing with the HH pattern waiting time. A run requires consecutive successes, unlike merely collecting

r

successes in total.